Asymptotic exponentiality of the tail of the waiting-time distribution in a Ph/Ph/C queue
- 1 September 1981
- journal article
- Published by Cambridge University Press (CUP) in Advances in Applied Probability
- Vol. 13 (3), 619-630
- https://doi.org/10.2307/1426788
Abstract
It is shown that, in a multiserver queue with interarrival and service-time distributions of phase type (PH/PH/c), the waiting-time distribution W(x) has an asymptotically exponential tail, i.e., 1 – W(x) ∽ Ke–ckx. The parameter k is the unique positive number satisfying T*(ck) S*(–k) = 1, where T*(s) and S*(s) are the Laplace–Stieltjes transforms of the interarrival and the service-time distributions. It is also shown that the queue-length distribution has an asymptotically geometric tail with the rate of decay η = T*(ck). The proofs of these results are based on the matrix-geometric form of the state probabilities of the system in the steady state.The equation for k shows interesting relations between single- and multiserver queues in the rates of decay of the tails of the waiting-time and the queue-length distributions.The parameters k and η can be easily computed by solving an algebraic equation. The multiplicative constant K is not so easy to compute. In order to obtain its numerical value we have to solve the balance equations or estimate it from simulation.Keywords
This publication has 9 references indexed in Scilit:
- Matrix-Geometric Solutions in Stochastic Models. An Algorithmic Approach.Journal of the American Statistical Association, 1982
- The probabilistic significance of the rate matrix in matrix-geometric invariant vectorsJournal of Applied Probability, 1980
- Renewal processes of phase typeNaval Research Logistics Quarterly, 1978
- Markov chains with applications in queueing theory, which have a matrix-geometric invariant probability vectorAdvances in Applied Probability, 1978
- A NUMERICAL METHOD FOR THE STEADY-STATE PROBABILITIES OF A G1/G/C QUEUING SYSTEM IN A GENERAL CLASSJournal of the Operations Research Society of Japan, 1976
- Stochastic Processes in Queueing TheoryPublished by Springer Nature ,1976
- On the algebra of queuesJournal of Applied Probability, 1966
- CorrectionsJournal of Applied Probability, 1966
- The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)The Mathematical Gazette, 1961